By definition we have \(\widetilde{\mathcal {M}}_{\mathrm {c},\kappa }= \mathcal {M}_{\mathrm {c}}\otimes {\mathbb {1}} \vee {\hbox{\rm Ad\,}}\widetilde{V}_{\mathrm {c},\kappa }({\mathbb {1}}\otimes \mathcal {M}_{\mathrm {c}})\) 
and \(\mathcal {M}_{\mathrm {c}}\)  is generated by the exponential of fields \(\phi (f)\otimes {\mathbb {1}}, {\mathbb {1}}\otimes \phi (g)\) , where
\(f,g\)  are real test functions with \({\rm supp}f, {\rm supp}g \subset W_{\mathrm {R}}\)  and \(\mathcal {H}_{\mathrm {c}}= \mathcal {H}_{\mathrm {r}}\otimes \mathcal {H}_{\mathrm {r}}\)  is understood.
First let us consider the \(\mathcal {M}_{\mathrm {c}}\otimes {\mathbb {1}}\)  component.
Following [1]}, we consider pairs of complex valued test functions \(f,g\)  such that \(\overline{f} = g\) .
The complex field which generates the wedge algebra in [1]} is given in our notation by
\(\phi _{\mathrm {c}}(f\oplus g) &=& a_{\mathrm {c}}^\dagger (f^+\oplus g^+) + a_{\mathrm {c}}(J_1(f^-\oplus g^-)) \\&:=& a_+^\dagger (f^+) + a_-^\dagger (g^+) + a_+(\overline{g^-}) + a_-(\overline{f^-}) \\&=& \phi (f)\otimes {\mathbb {1}} + {\mathbb {1}}\otimes i\phi (f) + \phi (g)\otimes {\mathbb {1}} + {\mathbb {1}}\otimes i\phi (g) \\&=& \phi (f + \overline{f})\otimes {\mathbb {1}} + {\mathbb {1}}\otimes i \phi (f - \overline{f}) \\&=& \phi (2\Re f)\otimes {\mathbb {1}} - {\mathbb {1}}\otimes \phi (2\Im f),\) 
